Math.Physics.md

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page	Math/Physics

Cross product

calculator

Law of Cosines

$$c=\sqrt{a^{2}+b^{2}-2 a b \cdot \cos \gamma}$$

Small angle approximations:

$$\sin\theta \approx \theta, \tan\theta \approx \theta, \cos\theta \approx 1-\frac{\theta^2}{2}$$ $$ \sin(u)\sin(v) = \frac{1}{2}[\cos(u-v) - \cos(u+v)] $$ $$ \cos(u)\cos(v) = \frac{1}{2}[\cos(u-v) + \cos(u+v)] $$

Physics prequisite

force to torque:

$$ \tau = r \times F = rF\sin\theta $$

friction torque(viscous) >Wiki:

$$ \tau = -k\dot{\theta} $$

弹性力（k 为弹簧的劲度系数，x 为振子偏离平衡位置的位移）： $$ F = -kx $$ 阻尼力（c 为阻尼系数，v 为振子速度）： $$ F = -cv $$

Potential energy

Springs: $\frac{1}{2}k(x-x_0)^2$

Gravity: $mgh$

Kinetic energy

Linear motion: $\frac{1}{2}mv^Tv$

Angular motion: $\frac{1}{2}\omega^TI\omega$

Inertia

$I$ is always symmetric. Diagonal if CoM aligns with primary axes.

For cylinder: $I_z = \frac{1}{2}MR^2$, $I_x=I_y=\frac{1}{4}MR^2+\frac{1}{12}ML^2$